properly included en · ADJ
Meanings
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(not-comparable) Being a proper subset; having the property that every element of a set A is also an element of a set B, and A is not equal to B.
If A is a subset of B but A ≠ B, then A is said to be properly included in B.
2003, Herbert B. Enderton, Elements of Set Theory, Academic Press, page 22:
Antonyms
improperly included · equal to